What's a word for when someone is wrong, and they know they're wrong, but they insist that they are right?

EDIT:
If someone does something and you know they did it, and they know they did it, but they keep telling everyone they didn't do it, and they go out of their way to try and convince others that they didn't do it.

As @jimreed says, such a person is a liar. This basically covers all contexts where someone says something that they know is untrue. But OP is specifically looking for a way to describe a liar who is aware that at least some other people know for certain that what he says untrue.

A common coarse slang term that comes to mind is bullshitter. I'm not saying Urban Dictionary is always a reliable authority, but I can't fault the example in definition 7 there - Someone who will claim he drank 2 litres of vodka and was not drunk.

+1 with caveat: personally I feel that BRAZEN is more fitting to what the OP is looking for (as in "stupidly bold"). To me, BAREFACED denotes that most people don't believe the lie. The lie is so obvious, it is "written on the liar's bare face".
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Martin S. StollerSep 26 '11 at 20:19

1

@Martin S. Stoller: I didn't say so in the answer, but I personally would probably normally call someone a brazen liar, and accuse them of coming out with a barefaced lie. If you'd habitually make that subtle distinction between them, you're a more precise speaker than me. I'd just use my two pairings practically every time because they're what rolls off my tongue. Which makes sense with barefaced lie, because I'd hear that more. I think I say brazen liar because I also say brazen hussy, so I'm more used to applying brazen to people, not to lies.
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FumbleFingersSep 27 '11 at 1:18

...plus whilst I like the idea of written on the liar's bare face, I don't think the *guilt written all over your face interpretation directly relates to the origin of barefaced, which is simply beardless, undisguised.
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FumbleFingersSep 27 '11 at 2:08

Have you noticed how variants on barefaced lie/liar have been cropping up of late? You now see boldfaced and even baldfaced showing up a good deal, mostly over the last 30–40 years. Ngrams here and here. Plus rerun as a single unhyphenated word for even more of a neologistic shift.
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tchristJul 29 '12 at 14:39